International audienceThe realist and interactive simulation of deformable objects has become a challenge in Computer Graphics. For this, the Tensor-Mass model is a good candidate: it enables local solving of mechanical equations, making it easier to control deformations from collisions or tool interaction. In this paper, a GPU implementation is presented for the implicit integration scheme. Results show a notable speedup, especially for complex scenes

Faure, X.

Jaillet, Fabrice

Moreau, J.-M.

Zara, F.

English

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Implicit Tensor-Mass solver on the GPU
X. Faure, F. Zara, Fabrice Jaillet, J.-M. Moreau
To cite this version:
X. Faure, F. Zara, Fabrice Jaillet, J.-M. Moreau. Implicit Tensor-Mass solver on the GPU. Eurograph-
ics/ ACM SIGGRAPH Symposium on Computer Animation (2012), Jul 2012, Lausanne, Switzerland.
pp.NA. ￿hal-00838722￿
Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2012) Posters and Demos
M. Lau (Posters Editor)
Implicit Tensor-Mass solver on the GPU
X. Faure1,2 and F. Zara2 and F. Jaillet2,3 and J-M. Moreau2
1Financed by the PRRH (Rhône-Alpes Research Program on Hadrontherapy) for ETOILE (National French Hadrontherapy Centre)
2Université de Lyon, CNRS, Université Lyon 1, LIRIS, SAARA team, UMR5205, F-69622, Villeurbanne
3Université de Lyon, IUT Lyon 1, Computer Science Department, F-01000, Bourg-en-Bresse
Figure 1: (1) Rendered beam for Hooke’s and Saint Venant-Kirchhoff’s material, and initial state (from left to right) - (2-4)
Deformation simulation of a rabbit (initial 3D mesh courtesy of L. Stanculescu).
Abstract
The realist and interactive simulation of deformable objects has become a challenge in Computer Graphics. For
this, the Tensor-Mass model is a good candidate: it enables local solving of mechanical equations, making it easier
to control deformations from collisions or tool interaction. In this paper, a GPU implementation is presented for
the implicit integration scheme. Results show a notable speedup, especially for complex scenes.
Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry,
Object Modeling—Physically based modeling I.6.8 [Simulation And Modeling]: Types of Simulation—Parallel
1. Introduction
The Tensor-Mass (TM) approach is a good candidate to
handle deformable objects within the context of physically-
based simulations in complex interactive scenes. The defor-
mation forces are derived from the FEM mechanical formu-
lation, but are next computed locally and iteratively for each
discretized element. Besides, this allows to handle more di-
rectly and easily topological changes (cutting) and external
interactions (collisions, tools), in the same way as Mass-
Spring Systems.
Hence, several formulations have been proposed to ac-
count for various mechanical behaviors: the linear Hookean
model [CDA00]; the non-linear geometrical model based
on Saint Venant-Kirchhoff’s elasticity model [PDA00],
or anisotropic material [Pic03]; or an extension with
pre-computation for non-linear visco-elastic deforma-
tions [SDR⇤05].
But, as far as we know, the problem of the parallel imple-
mentation of the TM model has not yet been addressed. In
this paper, a GPU implementation is presented, considering
both linear and non-linear mechanical behaviors and using
an implicit integration scheme to ensure the unconditional
stability of our simulation.
2. Simulation of the deformation of an object
The TM approach is based on the domain’s discretization
into several elements, and the mechanical equations are next
solved locally, involving the following main steps for each
element:
• Discretization of the displacement UE with the definition
of interpolation functions L according to the chosen type
of elements (hexahedron, tetrahedron, etc.);
• Computation of the strain-tensor according to the chosen
elasticity model amongst the most employed in interac-
tive simulations, namely a) Hooke’s or b) Saint Venant-
Kirchhoff’s elasticity models -within the element, at X :
a) el(X) = 12 (rUT (X)+rU(X))
b) enl(X) = 12 (rUT (X)+rU(X)+rUT (X)rU(X))
• Computation of the deformation energyWE , according to
the displacement of the element’s node;
c  The Eurographics Association 2012.
X. Faure & F. Zara & F. Jaillet & J-M. Moreau / Implicit Tensor-Mass solver on the GPU
• Computation of the elasticity force FE by the evaluation
of the derivative of the deformation energyWE .
3. Force computation on the GPU
The parallel computation of the forces on the GPU is divided
into 2 tasks, involving a set of kernels (Alg. 1):
1. First, we compute and store the forces applied on each
node of a given element. This computation (kernel1) is
made for each element of the domain.
2. Next, we sum these partial forces to obtain the total forces
applied on each node, involved by the different elements
of the domain. This computation (kernel2) is made for
each node of the domain.
Algorithm 1 Force computation on the GPU
1: {N : number of elements; m : total number of nodes}
2: {n : number of nodes per element}
3: {Nn : max number of neighbor elements for a node}
4: // Task 1- Computation of partial forces
5: for e = 0 to N 1 do
6: // Execution of N kernel1
7: for v = 0 to n 1 do
8: PartialForce[ForceIndex[e][v]][index[e][v]]=Force(...);
9: end for
10: end for
11: // Task 2 - Sum of partial forces
12: for i = 0 to m do
13: // Execution of m kernel2
14: for j = 0 to Nn 1 do
15: TotalForce[i] += PartialForce[i][j];
16: end for
17: end for
Moreover, specific data structures are defined for a dis-
cretization of the domain intoN elements involvingm nodes:
• index (of size N⇥n) stores the relationship between lo-
cal and global indexation for each node of each element.
• PartialForce (of size Nn⇥m⇥3) enables the storage
of 3D coordinates of partial forces for each node consid-
ering its global indexation.
• TotalForce (of sizem⇥3) stores the sum of the partial
forces for each node considering its global indexation.
• ForceIndex (of size N ⇥ n) stores the index of data
structure PartialForce.
4. Implicit integration method on the GPU
Then, the dynamical equations are derived from Newton’s
laws to compute the object acceleration. An implicit in-
tegration scheme, requiring computation of the derivatives
of forces, is then used to obtain deformation and displace-
ment. The parallel algorithm for this is similar to Alg. 1 -
with the only difference in the use of DForce() instead of
Force(), which depends on the same parameters but also
on the time step h and the current nodes velocity, to directly
compute h2 ∂F∂U V (t) or h
∂F
∂U DV , and avoid the storage of the
sparse Jacobian matrix ∂F∂U . Next, the linear system is solved
on the GPU with the Conjugate Gradient method, follow-
ing [ACF11].
5. Results and performances
We compare running times on the CPU and GPU, using
Linux Ubuntu11.04. The CPU is an Intel R  Xeon R  4 cores
@3.07 GHz. The GPU is a GeForce GTX 560, 2047 MB,
56 cores @1.620 GHz. Fig. 2 presents the speedup (ratio be-
tween GPU and CPU execution time) obtained for triangles
and tetrahedra for both the linear and non-linear TM model
(Fig. 1). Moreover, similar results are presented for SOFA’s
FEM implementation, i.e. the corotational FEM model for
tetrahedra [NP05], in its GPU version according to [ACF11].
A speedup of 25.5 is reached for SOFA’s FEM implemen-
tation and of 29.5 for our TM, for a beam composed of
307,200 elements.
 0
 5
 10
 15
 20
 25
 30
 0  50000  100000  150000  200000  250000  300000  350000
Sp
ee
du
p
Number of elements
Linear Triangle TM
Non Linear Triangle TM
Linear Tetra TM
Non Linear Tetra TM
Rotational Tetra FEM
Figure 2: Performances of the TM parallelization.
6. Conclusion and perspectives
In this paper, we presented an original implementation of the
TM model on the GPU that considerably speeds up the sim-
ulation times. Comparisons between running times on the
CPU and the GPU suggest that the parallel implementation
of the model becomes interesting for increasingly complex
computations.
References
[ACF11] ALLARD J., COURTECUISSE H., FAURE F.: Implicit
FEM Solver on GPU for Interactive Deformation Simulation.
In GPU Computing Gems Jade Edition. NVIDIA/Elsevier, Sept.
2011, ch. 21. 2
[CDA00] COTIN S., DELINGETTE H., AYACHE N.: A hybrid
elastic model for real-time cutting, deformations, and force feed-
back for surgery training and simulation. The Visual Computer
16, 8 (2000), 437–452. 1
[NP05] NESME M., PAYAN Y.: Efficient, physically plausible fi-
nite elements. Eurographics (short papers) (2005), 1–4. 2
[PDA00] PICINBONO G., DELINGETTE H., AYACHE N.: Real-
Time Large Displacement Elasticity for Surgery Simulation:
Non-linear Tensor-Mass Model. In Proceedings of MICCAI’00
(London, UK, 2000), Springer-Verlag, pp. 643–652. 1
[Pic03] PICINBONO G.: Non-linear anisotropic elasticity for real-
time surgery simulation. Graphical Models 65, 5 (Sept. 2003),
305–321. 1
[SDR⇤05] SCHWARTZ J., DENNINGER M., RANCOURT D.,
MOISAN C., LAURENDEAU D.: Modelling liver tissue proper-
ties using a non-linear visco-elastic model for surgery simulation.
Medical Image Analysis 9, 2 (2005), 103–112. 1
c  The Eurographics Association 2012.


Implicit Tensor-Mass solver on the GPU

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